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Continuous Probability Distributions
A continuous probability distribution is a representation of a variable that can take a continuous range of values.
Learning Objective

Explain probability density function in continuous probability distribution
Key Points
 A probability density function is a function that describes the relative likelihood for a random variable to take on a given value.
 Intuitively, a continuous random variable is the one which can take a continuous range of values — as opposed to a discrete distribution, where the set of possible values for the random variable is at most countable.
 While for a discrete distribution an event with probability zero is impossible (e.g. rolling 3½ on a standard die is impossible, and has probability zero), this is not so in the case of a continuous random variable.
Term

Lebesgue measure
The unique complete translationinvariant measure for the σalgebra which contains all kcells—in and which assigns a measure to each kcell equal to that kcell's volume (as defined in Euclidean geometry: i.e., the volume of the kcell equals the product of the lengths of its sides).
Full Text
A continuous probability distribution is a probability distribution that has a probability density function. Mathematicians also call such a distribution "absolutely continuous," since its cumulative distribution function is absolutely continuous with respect to the Lebesgue measure λ. If the distribution of X is continuous, then X is called a continuous random variable. There are many examples of continuous probability distributions: normal, uniform, chisquared, and others.
Intuitively, a continuous random variable is the one which can take a continuous range of values—as opposed to a discrete distribution, in which the set of possible values for the random variable is at most countable. While for a discrete distribution an event with probability zero is impossible (e.g. rolling 3½ on a standard die is impossible, and has probability zero), this is not so in the case of a continuous random variable.
For example, if one measures the width of an oak leaf, the result of 3½ cm is possible; however, it has probability zero because there are uncountably many other potential values even between 3 cm and 4 cm. Each of these individual outcomes has probability zero, yet the probability that the outcome will fall into the interval (3 cm, 4 cm) is nonzero. This apparent paradox is resolved given that the probability that X attains some value within an infinite set, such as an interval, cannot be found by naively adding the probabilities for individual values. Formally, each value has an infinitesimally small probability, which statistically is equivalent to zero.
The definition states that a continuous probability distribution must possess a density; or equivalently, its cumulative distribution function be absolutely continuous. This requirement is stronger than simple continuity of the cumulative distribution function, and there is a special class of distributions—singular distributions, which are neither continuous nor discrete nor a mixture of those. An example is given by the Cantor distribution. Such singular distributions, however, are never encountered in practice.
Probability Density Functions
In theory, a probability density function is a function that describes the relative likelihood for a random variable to take on a given value. The probability for the random variable to fall within a particular region is given by the integral of this variable's density over the region. The probability density function is nonnegative everywhere, and its integral over the entire space is equal to one.
Unlike a probability, a probability density function can take on values greater than one. For example, the uniform distribution on the interval [0, ½] has probability density f(x) = 2 for 0 ≤ x ≤ ½ and f(x) = 0 elsewhere. The standard normal distribution has probability density .
Boxplot Versus Probability Density Function
Boxplot and probability density function of a normal distribution N(0, 2).
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Key Term Reference
 continuous random variable
 Appears in these related concepts: Expected Value and Standard Error and The Correction Factor
 cumulative distribution function
 Appears in these related concepts: The Uniform Distribution, Significance Levels, and Determining Sample Size
 density
 Appears in these related concepts: The Density Scale, Quorum Sensing, and Water’s States: Gas, Liquid, and Solid
 distribution
 Appears in these related concepts: Application of Knowledge, Monte Carlo Simulation, and Selling to Consumers
 integral
 Appears in these related concepts: Tangent and Velocity Problems, Indefinite Integrals and the Net Change Theorem, and Volumes
 normal distribution
 Appears in these related concepts: Shapes of Sampling Distributions, The Average and the Histogram, and Standard Deviation: Definition and Calculation
 probability
 Appears in these related concepts: The Addition Rule, Theoretical Probability, and Rules of Probability for Mendelian Inheritance
 probability density function
 Appears in these related concepts: Probability, Continuous Sampling Distributions, and Philosophical Implications
 probability distribution
 Appears in these related concepts: Probability Distributions for Discrete Random Variables, Recognizing and Using a Histogram, and Overview of How to Assess StandAlone Risk
 random variable
 Appears in these related concepts: Chance Processes, Two Types of Random Variables, and Expected Value
 range
 Appears in these related concepts: Range, The Derivative as a Function, and Visualizing Domain and Range
 standard normal distribution
 Appears in these related concepts: The Gauss Model, The Normal Distribution, and The Standard Normal Curve
 uniform distribution
 Appears in these related concepts: Which Average: Mean, Mode, or Median?, Example: Test for Goodness of Fit, and SinglePopulation Inferences
 variable
 Appears in these related concepts: Fundamentals of Statistics, The Linear Function f(x) = mx + b and Slope, and Math Review
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Cite This Source
Source: Boundless. “Continuous Probability Distributions.” Boundless Statistics. Boundless, 21 Jul. 2015. Retrieved 01 Dec. 2015 from https://www.boundless.com/statistics/textbooks/boundlessstatisticstextbook/continuousrandomvariables10/thenormalcurve39/continuousprobabilitydistributions1872635/